How to Identify Equivalent Equations

Oct 13, 2025 by miodeya.com 47 views

Understanding the Problem

The given equation is $2x^2 - 32 = -12x$ . To find an equivalent equation, we need to manipulate the given equation to match one of the options provided. This involves rearranging the terms and possibly adding or subtracting constants to make the equation match the desired form.

Step 1: Rearrange the Given Equation

The first step is to isolate the quadratic term on one side of the equation. We can do this by adding $12x$ to both sides of the equation:

2x^2 - 32 + 12x = -12x + 12x

This simplifies to:

2x^2 + 12x - 32 = 0

Step 2: Compare with the Options

Now that we have the rearranged equation, we can compare it with the options provided. The equation we obtained is:

2x^2 + 12x - 32 = 0

This matches option D.

Conclusion

Based on the rearrangement of the given equation, we found that the equivalent equation is:

2x^2 + 12x - 32 = 0

This matches option D. Therefore, the correct answer is D.

Why is this the Correct Answer?

The correct answer is D because we rearranged the given equation to match the desired form, and the resulting equation matches option D. This demonstrates that the given equation is equivalent to the equation in option D.

What is the Importance of Equivalent Equations?

Equivalent equations are important in mathematics because they allow us to simplify complex equations and make them easier to solve. By rearranging the terms and possibly adding or subtracting constants, we can transform an equation into a more manageable form. This can be particularly useful when solving quadratic equations, as it can help us find the solutions more easily.

How to Identify Equivalent Equations

To identify equivalent equations, we need to look for equations that have the same solutions. This means that the equations must have the same values of x that satisfy the equation. We can check for equivalent equations by rearranging the terms and possibly adding or subtracting constants. If the resulting equation matches one of the options, then the equations are equivalent.

Common Mistakes to Avoid

When working with equivalent equations, there are several common mistakes to avoid. These include:

Not rearranging the terms correctly: Make sure to add or subtract the correct constants to isolate the quadratic term.
Not comparing the equations correctly: Make sure to compare the equations carefully to ensure that they have the same solutions.
Not checking for equivalent equations: Make sure to check for equivalent equations by rearranging the terms and possibly adding or subtracting constants.

Real-World Applications

Equivalent equations have many real-world applications. For example, in physics, equivalent equations can be used to describe the motion of objects. In engineering, equivalent equations can be used to design and optimize systems. In economics, equivalent equations can be used to model and analyze economic systems.

Conclusion

Q: What is an equivalent equation?

A: An equivalent equation is an equation that has the same solutions as another equation. This means that the equations must have the same values of x that satisfy the equation.

Q: How do I identify equivalent equations?

A: To identify equivalent equations, you need to look for equations that have the same solutions. This means that the equations must have the same values of x that satisfy the equation. You can check for equivalent equations by rearranging the terms and possibly adding or subtracting constants.

Q: What are some common mistakes to avoid when working with equivalent equations?

A: Some common mistakes to avoid when working with equivalent equations include:

Not rearranging the terms correctly
Not comparing the equations correctly
Not checking for equivalent equations

Q: How do I rearrange the terms of an equation to make it equivalent to another equation?

A: To rearrange the terms of an equation, you need to add or subtract the same constant to both sides of the equation. This will help you isolate the quadratic term and make the equation more manageable.

Q: What is the importance of equivalent equations in real-world applications?

A: Equivalent equations have many real-world applications. For example, in physics, equivalent equations can be used to describe the motion of objects. In engineering, equivalent equations can be used to design and optimize systems. In economics, equivalent equations can be used to model and analyze economic systems.

Q: Can you give an example of how to use equivalent equations in a real-world application?

A: Here's an example of how to use equivalent equations in a real-world application:

Suppose you are designing a bridge and you want to know the maximum weight that the bridge can hold. You can use an equivalent equation to model the weight of the bridge and the weight of the cars that will be driving on it. By rearranging the terms of the equation, you can find the maximum weight that the bridge can hold.

Q: How do I know if two equations are equivalent?

A: To know if two equations are equivalent, you need to check if they have the same solutions. You can do this by substituting the values of x into both equations and checking if they are true.

Q: Can you give an example of how to check if two equations are equivalent?

A: Here's an example of how to check if two equations are equivalent:

Suppose you have two equations:

Equation 1: 2x^2 + 12x - 32 = 0 Equation 2: 2x^2 + 12x - 32 + 5 = 0

To check if these equations are equivalent, you can substitute the values of x into both equations and check if they are true. If they are true, then the equations are equivalent.

Q: What is the difference between an equivalent equation and a similar equation?

A: An equivalent equation is an equation that has the same solutions as another equation. A similar equation is an equation that has a similar form to another equation, but may not have the same solutions.

Q: Can you give an example of how to use similar equations in a real-world application?

A: Here's an example of how to use similar equations in a real-world application:

Suppose you are designing a system and you want to know the maximum capacity of the system. You can use a similar equation to model the capacity of the system and the demand for the system. By rearranging the terms of the equation, you can find the maximum capacity of the system.

Conclusion

In conclusion, equivalent equations are an important concept in mathematics that allows us to simplify complex equations and make them easier to solve. By rearranging the terms and possibly adding or subtracting constants, we can transform an equation into a more manageable form. This can be particularly useful when solving quadratic equations, as it can help us find the solutions more easily.